Multiplying Binomials Calculator - FOIL Expansion of (ax + b)(cx + d)

A multiplying binomials calculator is a browser-based tool that takes two binomials in the form (ax + b) and (cx + d) and expands their product into a quadratic Ax^2 + Bx + C, with the four FOIL partial products shown alongside.

• • Algebra homework and FOIL checks: Verify a hand-written expansion of two binomials and confirm the four FOIL partial products before combining them into a single quadratic.
• • Difference of squares and perfect squares: Recognize special forms such as (x - 7)(x + 7) = x^2 - 49 and (3x - 2)^2 = 9x^2 - 12x + 4 from the combined coefficients.
• • Negative and fractional coefficients: Expand binomials that include negative terms or fractional coefficients and read the sign-corrected partial products.
• • Pre-factoring setup: Use the expanded quadratic to set up factoring, since (ax + b)(cx + d) is the standard form a quadratic is factored back into.

The calculator exposes four input fields, two for each binomial, with the first binomial's coefficient and constant in the top row and the second binomial's in the bottom row. The result panel then shows the four FOIL partial products and the combined A, B, C coefficients that a textbook would expect.

Once the FOIL expansion is in hand, the Factoring Trinomials Calculator runs the reverse step and factors the same Ax^2 + Bx + C result back into two binomials.

The calculator reads the four coefficient inputs, multiplies the First, Outer, Inner, and Last pairs, and combines the Outer and Inner partial products into the middle coefficient.

• a: Coefficient of x in the first binomial.
• b: Constant term of the first binomial.
• c: Coefficient of x in the second binomial.
• d: Constant term of the second binomial.
• A: x^2 coefficient, equal to a times c.
• B: x coefficient, equal to a*d + b*c.
• C: Constant term, equal to b times d.

Every recalculation runs in the browser on each input event, so the four FOIL partial products and the combined coefficients update without a page reload. Setting a, b, c, or d to 0 produces a degenerate case such as a constant-only binomial, and the formula still works: the x^2 coefficient becomes 0 and the result is linear.

According to Wikipedia, A binomial is a polynomial that is the sum of two terms, and multiplying two binomials uses the distributive property to combine four partial products: First (a·c), Outer (a·d), Inner (b·c), and Last (b·d).

When the user wants to divide the expanded quadratic by one of the original binomials, the Polynomial Division Calculator performs long division and returns the remaining linear factor as a quotient and a remainder.

Four small ideas cover every binomial product, from a single FOIL step to the special forms a textbook often tests.

These four ideas are why a single FOIL step works for two-binomial products no matter what the coefficients are. The Outer and Inner partial products can be negative, fractional, or zero, and the calculator handles the sign and fraction rules automatically.

Once the FOIL step has produced an expanded quadratic, the Add Subtract Polynomials Calculator combines it with another polynomial by adding or subtracting matching terms.

Type the two binomials, read the four FOIL partial products, and check the combined quadratic to confirm the expansion.

1. 1 Enter the first binomial: Type the coefficient of x in the first binomial in field a, and the constant term in field b. Use a negative constant to enter a difference.
2. 2 Enter the second binomial: Type the coefficient of x in the second binomial in field c, and the constant term in field d. Set c to 0 if the second binomial is constant-only.
3. 3 Read the four FOIL partial products: Look at the result panel. The First, Outer, Inner, and Last rows show each partial product individually.
4. 4 Read the combined quadratic: Use the A, B, and C rows to read the expanded quadratic in the form A·x^2 + B·x + C. The calculator shows A in the primary result tile, with B and C underneath.
5. 5 Verify special forms: When B is 0, the result is a difference of squares. When the two binomials are identical, the Outer and Inner partial products are equal.
6. 6 Reset to start over: Press Reset to restore the default binomials, which is useful when working through a list of FOIL problems.

Try the calculator with a = 3, b = -2, c = 3, d = -2. The four partial products are 9, -6, -6, and 4, and the combined quadratic is 9x^2 - 12x + 4, which is the perfect-square trinomial (3x - 2)^2.

When the combined quadratic needs its zeros, the Quadratic Formula Calculator plugs A, B, and C into the quadratic formula and returns the two roots.

The tool gives you the four FOIL partial products and the combined quadratic in the same view, so the FOIL step and the final answer never get separated.

• • FOIL step made visible: Each partial product is reported in its own row, so you can read the First, Outer, Inner, and Last values without doing the mental arithmetic twice.
• • Combined quadratic alongside the steps: The A, B, and C coefficients sit next to the four partial products, so you can copy the final quadratic into a graphing or factoring step.
• • Negative and fractional coefficients: Negative constants and fractional coefficients are handled in a single pass, so FOIL works for (2x - 5)(4x + 3) and (0.5x + 1)(2x - 4) the same way.
• • Special-product detection: A B value of 0 signals a difference of squares, and equal Outer and Inner values signal a perfect-square trinomial.
• • Real-time recalculation: Every keystroke updates the result panel, so you can iterate over a list of FOIL problems without pressing a button.

Because the layout reports the four partial products and the combined quadratic side by side, the calculator doubles as a FOIL verification tool. Read the partial product rows to confirm the FOIL step, then read the A, B, and C rows to confirm the combined quadratic, all in the same panel.

Once the expanded quadratic is in hand, the Polynomial Graphing Calculator plots A·x^2 + B·x + C so the user can see where the parabola crosses the x-axis.

Four input coefficients shape the result, and a small set of caveats keeps the FOIL expansion honest for negative, fractional, and degenerate cases.

• • The calculator accepts only the (ax + b)(cx + d) form, not three-term polynomials. To expand a trinomial times a binomial, run the FOIL step twice and combine the partial products by hand, or use a general polynomial multiplication tool.
• • Inputs are real numbers, so coefficients like x^2 inside a binomial are not supported. The form is strictly two terms per factor, which is the standard binomial definition in algebra.
• • The result is shown as the combined quadratic A·x^2 + B·x + C with all four FOIL partial products. The tool does not solve for x; it only expands the product. Use a quadratic formula or factoring tool to find the roots of the result.

Treat the result as the same expansion you would get on paper, with three caveats: three-term polynomials are out of scope, the form is strictly two-term binomials, and the result is an expanded quadratic rather than a list of roots.

According to Omni Calculator, Multiplying binomials expands (ax + b)(cx + d) into a·c·x^2 + (a·d + b·c)·x + b·d, with the FOIL method naming the First, Outer, Inner, and Last partial products.

According to Math is Fun, Every term in the first polynomial must be multiplied by every term in the second, and FOIL is a quick way to remember the four products for two binomials.

When the FOIL result needs to be combined with another number, the Multiplication Calculator multiplies the combined quadratic as a scalar against a single factor in a single step.